Wavemaker

11. Wavemaker#

WAVE_MAKER is useful for wave-resolving simulations in the nearshore zone. In wave-resolving nearshore problem, a spectrum of linear waves can be forced at the offshore boundary via an analytical wavemaker (parameters set in croco.in and used in wave_maker.h as part of the ana_bry subroutine). The spectrum is based by default on a JONSWAP statistical distribution and has frequency and directional spreading.

A single-sum wavemaker (i.e. each component of the wave has a distinct frequency and direction) is implemented in CROCO to alleviate the problem of coherent wave interference arising from double-summation methods (Treillou et al. [2024]). A single-sum wavemaker solves this problem and is also much more cost-effective (and does not require high-frequency resolution).

11.1. Formulation#

The offshore wavemaker is used with OBC_WEST or OBC_EAST, providing BRY array values for free surface elevation \(\eta\), cross-shore and alongshore velocities \((u,v)\), according to the following equations:

\[ \begin{align}\begin{aligned}\eta_{bc}(y,t)=\sum_i^{N\times M} a_i \cos(k_{y,i}y - \omega_i t - \phi_{i})\\u_{bc}(x,y,t)=\eta_{bc}(y,t) \omega_p \cos(\theta_m) \frac{\cosh(k_p(z+h))}{\sinh(k_p h)}\\v_{bc}(x,y,t)=\eta_{bc}(y,t) \omega_p \sin(\theta_m) \frac{\cosh(k_p(z+h))}{\sinh(k_p h)}\end{aligned}\end{align} \]

with :

  • \(a_i = a_m \sqrt{\frac{D(\theta_i) S(\omega_i) d\omega}{\sum_i D(\theta_i) S(\omega_i) d\omega}}\) : amplitude given by the statistical distribution \(S(\omega_i,\theta_i)=S(\omega_i) \times D(\theta_i)\), the product of a JONSWAP frequency spectrum \(S(\omega_i)\) and a Gaussian-type directional spectrum \(D(\theta_i)\). \(i\) is the index of spectral distribution in frequency and direction; N and M are the number of frequencies and directions; and \(d\omega\) is frequency resolution.

  • \(a_m=\sqrt{H_s/8}\) is the rms wave amplitude (related to significant wave height \(H_s\)).

  • \(\theta_i, \theta_m, \sigma_{\theta}\) : i-th wave direction, mean wave direction, and directional spread around the mean.

  • \(k_{y,i}=k_i \text{sin}(\theta_i )\) : alongshore wavenumber (\(k_i\) being the linear theory wavenumber given by \(\omega_i^2 = g k_i \text{tanh}(k_i h)\) with \(h\) the mean water depth)

  • \(\omega_p, k_p\) : peak frequency and wavenumber,

  • \(\phi_i\) uniformly distributed random phase.

11.2. Input parameters#

Input parameters for the wavemaker are set in croco.in under keyword: wave_maker (e.g., Flash Rips test case):

wave_maker:  amp [m], prd [s], ang [deg], ang spread [deg], freq spread (gamma)
             0.24     13.       0.         30.               3.3

amp: wave amplitude \(a_m=\sqrt{H_s/8}\)

prd: peak period \(T_p = 2 \pi / \omega_p\)

ang: mean wave angle \(\theta_m\)

ang spread: directional spread \(\sigma_{\theta}\)

freq spread: peak enhancement factor \(\gamma\) of JONSWAP spectrum

Note

The directional spread is essential for generating short-crested waves (and flash rips). The mean crest length should vary as \(\lambda_p/(2 \sin{\sigma_{\theta}})\), with \(\lambda_p=2 \pi / k_p\) the peak wavelength.

Note

if WAVE_MAKER_SPECTRUM is not defined (see below), the wavemaker produces a monochromatic wave.

Related CPP options:

WAVE_MAKER

Activate Wavemaker for wave resolving nonhydrostatic case

WAVE_MAKER_SPECTRUM

Activate Wave spectrum with frequency spread (default: JONSWAP)

WAVE_MAKER_DSPREAD

Add directional spread in wave spectrum

WAVE_MAKER_OBLIQUE

Specify if Non-zero mean angle (compute v component)

Preselected options:

#ifdef WAVE_MAKER
# if defined WAVE_MAKER_JONSWAP || defined WAVE_MAKER_GAUSSIAN
#  define WAVE_MAKER_SPECTRUM
# endif
# ifdef WAVE_MAKER_SPECTRUM
#  ifdef WAVE_MAKER_JONSWAP
#  elif defined WAVE_MAKER_GAUSSIAN
#  else
#   define WAVE_MAKER_JONSWAP /* default */
#  endif
# endif
# if defined WAVE_MAKER_DSPREAD && defined NS_PERIODIC
#  define WAVE_MAKER_DSPREAD_PER /* correct direction for periodicity */
# endif
# ifndef WAVE_MAKER_SPECTRUM
#  define STOKES_WAVES
# endif
#endif

11.3. JONSWAP spectrum#

As previously said, the spectrum distribution is the product of a JONSWAP frequency spectrum \(S(\omega)\) and a Gaussian-type directional spectrum \(D(\theta)\) : \(S(\omega,\theta)=S(\omega) \times D(\theta)\).

The JONSWAP spectrum is formulated in CROCO as follows:

\[S(\omega) = H_s^2 \beta_J \omega_p^4 \omega^{-5} \text{exp} [-1.25 \omega^{-4}\omega_p^4] \gamma^r\]

function of the significant wave height \(H_s\), peak wave frequency \(\omega_p\), peak enhancement factor \(\gamma\), and:

\[ \begin{align}\begin{aligned}r = \exp{ \left [ -\frac{1}{2} \left ( \frac{\omega - \omega_p}{\sigma_{\omega} \omega_p} \right )^2 \right ]}\\\begin{split}\sigma_{\omega} = 0.07, & \text{ if}\ \omega \leq \omega_p \\ 0.09, & \text{ if}\ \omega>\omega_p\end{split}\\\beta_J = \frac{0.06238 (1.094 - 0.01915 \log{\gamma})}{0.23+0.0336\gamma - 0.185(1.9+\gamma)^{-1}}\end{aligned}\end{align} \]

The directional spectrum around the mean direction \(\theta_m\), with directional spread \(\sigma_{\theta}\), is:

\[D(\theta) = \exp{\left[-\left(\frac{\theta - \theta_m}{1.5 \sigma_{\theta}}\right)^2\right]}\]

with \(\int_{\theta_{min}}^{\theta_{max}} D(\theta) d\theta = 1\) to ensure that the directional spread (\(\sigma_{\theta}\)) around the mean wave angle does not affect wave energy.

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